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Increasing variability always degrades the performance of a production system.

Â This is the law of variability.

Â This law, as discussed by Hopp and Spearmen,

Â shows that understanding variation is very important for the quality of a process.

Â Therefore, I want to show you how to study variability across groups.

Â I will explain how to perform a test for equal variances.

Â Moreover, I will motivate why this is useful.

Â Let's go back to the example of four machines producing coffee.

Â Together, we investigated that the average moisture content

Â of the coffee beans depended on the machine it was produced on.

Â The data looked like this.

Â We showed that the means of moisture percentage differ across the four machines.

Â But what about the variation within a machine?

Â Machine two has a wide range of moisture contents,

Â while machine four appeared a lot more consistent.

Â For producing consistent coffee,

Â we probably want the machine with the smallest variation.

Â Let's take a closer look at these two machines.

Â The mean level is near equal for machine two and four.

Â However, the variation of the two machines may differ.

Â This is the data of machine two in a time order.

Â And here, we see the data for a machine four in a time order.

Â The moisture content of machine two goes up and down more than that of machine four.

Â If the machine produces very inconsistently,

Â meaning a lot of variation,

Â this might cause problems.

Â It could lead to a lot of scrap and defects which will cost you money.

Â If you want your moisture percentage to be, for instance,

Â lower than 10.5 percent,

Â and there is a lot of variation,

Â like in machine two,

Â a lot of your batches will show a moisture percentage which is higher than this 10.5,

Â and these batches will have to be thrown away or reprocessed.

Â If the variation is smaller,

Â like in machine four,

Â this will happen less often.

Â Therefore, you will probably like to know if one

Â of the machine produces more consistently than the others.

Â The output of the ANOVA analysis already gave us some information on the variation.

Â We see that machine four has a smaller standard deviation and therefore,

Â produces most consistently within the sample at least.

Â However, are these differences between

Â these standard deviation is statistically significant,

Â or is it a coincidence that machine four has the smallest variation?

Â Let us use a statistical test,

Â the test of equality of variances to study this.

Â Now, pause your video,

Â load the data into Minitab before continuing.

Â This is what your data in Minitab should look like.

Â Note that I stacked the four columns with the individual machine data

Â into one column which is moisture and one column which shows the machine.

Â For testing of equal variances,

Â we have to go to the stats menu because this is statistical analysis.

Â On the stat, we go to ANOVA.

Â And there, you can find the test for equal variances here.

Â Now, Minitab asks you,

Â "What is your response?"

Â Well, our response or the y variable or the CTQ is moisture of course.

Â Next, we have to fill in the factors or the influence factors,

Â and that, in this example,

Â is of course machine.

Â Well, that's it.

Â So, okay.

Â Now, Minitab gives us some output.

Â And one of them is this graph,

Â the test of equal variances.

Â And the other output is in the session window,

Â lots of different, what, results.

Â Now, let's study the output.

Â The estimates for the standard deviation in

Â moisture percentage for the machines are given.

Â To check if the differences between these are statistically significant,

Â we look at the p-value,

Â compute it according to Levene's method.

Â In this case, the p-value is relatively large.

Â It shows us that there is a 32.5 percent

Â chance that the difference between the machines is a chance fluctuation,

Â which is much larger than our threshold of five percent.

Â This means that we did not find evidence that

Â there is a difference in variation between machines,

Â and we cannot draw any conclusions on what the machine produces most consistently.

Â Now, suppose that we would have had 50 measurements for each machine instead of 10.

Â This would give us a different dataset.

Â The individual value plot would look like this.

Â The equal variance test now gives us this output.

Â What would you conclude?

Â Do you see a statistical difference between the variances of the machines?

Â We see a very low p-value,

Â which means that there is a statistical difference between the variances of the machine.

Â And we can conclude the machine four produces most consistently.

Â The test for equal variances also has implications for the ANOVA analysis.

Â Remember that when we performed ANOVA analysis,

Â we selected the options and uncheck the box assume equal variances.

Â In this case of our first example,

Â we found no significant difference between the variances of each machine.

Â So, actually, we did not need to un-check this box.

Â It is preferred not to do this because then the p-value would be more precise.

Â Let's summarize. You now know why variances are

Â important and learned how to compare them across groups.

Â You know that knowledge about equality of

Â variances can be used to improve the p-value over and over.

Â